8 LEBESGUE'S CONVERGENCE THEOREMS 127 The usefulness of this theorem is that it does not require the convergence to be uniform (not even on compact subsets). In particular: (13.8.5) Let (/„) be a sequence of integrable functions^ such that the series with general term \fn\ djji is convergent. Then the series with general term fn(x) is 00 almost everywhere absolutely convergent in R, and if weputf(x) = £ fn(x\ the function f (defined almost everywhere) is integrable and we have (13.8.5.1) j/(x)^W = n|;J/B(x (" term-by-term integration of a series of integrable functions"). N For each integer N ^ 0, let #N = £ \fn\. The sequence (#N) is an increasing w=l sequence of integrable functions, and N r \L\ dn ((13.7.4) and (13.7.3)). By (13.8.1), the function # = lim #N = £ |/J is N->oo n-1 integrable. This implies first of all that g is finite almost everywhere (13.6.4), in other words that the series with general term \fn(x)\ is convergent almost everywhere, which proves the first assertion ((5.3.1) and (5.3.2)). Moreover, N for all N we have ]T \fn\ ^ g, from which by (13.8.4) follow the integrability n=l of /and the relation (13.8.5.1). (13.8.6) (i) Let E be a metric space, (x, z)\-^f(x, z) a mapping ofX x E into K, and z0 a point ofE. Suppose that (1) for each z e E, the function x\-+f(x, z) is integrable ; (2) for almost all x e X, the function z\-^>f(x, z) is continuous at ZQ ; (3) there exists an integrable function g ^ 0 such that., for all z e E, we have \f(x, z)\ ^ g(x) almost everywhere in X. Then h(z) = \f(x, z) dfi(x) is continuous at ZQ . (ii) Suppose furthermore that E is an open interval in R and that f satisfies the following conditions: (4) for almost all xeX, the function ZH->/(X, z) is finite and admits a derivative D2/(x, z);re, for alln. Then f is integrable and